Addresses @teknium1's review of #61950:
- The desktop live overlay (workspace-groups.ts) matched cwd membership
case-sensitively, so a fresh mixed-case/separator Windows session missed
its explicit/auto project until the next backend tree refresh. Mirror the
backend identity (isWindowsPath/comparisonSegments/pathKey) in isPathUnder,
liveSessionProjectId, and overlayRepoLanes lane matching. Comparison-only —
emitted ids/labels keep their spelling. POSIX stays case-sensitive.
- Backend _is_windows_path missed root-relative `\wsl.localhost\...` (single
leading backslash), leaving that historical spelling case-sensitive. Classify
any backslash-rooted path as Windows.
Tests: WSL-spelling collapse + explicit-project precedence (project_tree),
Windows/WSL live-overlay membership + POSIX case-sensitivity (workspace-groups).
